Incorrect sorting Formula

　

1. Recursive Method derivation of the error sorting Formula

When N numbered elements are placed in N numbered positions, the number of methods that do not correspond to each element number is represented by M (N), then M (n-1) it indicates that n-1 number elements are placed at N-1 numbers, and the number of methods does not correspond to each other, and so on.

Step 1: place the nth element in a single position, such as position K. There are n-1 methods in total;

Step 2: place the element number K. In this case, place it in the position N. For N-1 1 elements, since the K element to the position N, the remaining N-2 element has m (n-2) method; (2) the K element does not put it to the position N, then, for the n-1 elements, there are m (n-1) methods;

In summary

　　M (n) = (n-1) [M (n-2) + M (n-1)]

In particular, M (1) = 0, M (2) = 1

The following formula is derived from this recursive relationship:

For convenience, Set M (K) = k! N (k), (k = 1, 2 ,..., N)

N (1) = 0, N (2) = 1/2

N> = 3, n! N (n) = (n-1) (n-1 )! N (n-1) + (n-1 )! N (n-2)

That is, NN (n) = (n-1) n (n-1) + N (n-2)

So there is n (n)-N (n-1) =-[n (n-1)-N (n-2)]/n = (-1/n) [-1/(n-1)] [-1/(n-2)]… (-1/3) [N (2)-N (1)] = (-1) ^ n/n!

Therefore

N (n-1)-N (n-2) = (-1) ^ (n-1)/(n-1 )!

N (2)-N (1) = (-1) ^ 2/2!

Add to get

N (n) = (-1) ^ 2/2! +... + (-1) ^ (n-1)/(n-1 )! + (-1) ^ n/n!

Therefore

M (n) = n! [(-1) ^ 2/2! +... + (-1) ^ (n-1)/(n-1 )! + (-1) ^ n/n!]

You can get

The error row formula is m (n) = n! (1/2! -1/3! + ..... + (-1) ^ n/n !)

 

2. Refresh Principle

Positive integers 1, 2, 3 ,...... And n are arranged in full order with n! (N-1 )!, When K is 1, 2, 3 ,...... , N, there are N * (n-1 )! Sorting, because it is a wrong arrangement, these arrangements should be excluded, but at this time there are two good rows of the Arrangement of multiple rows in addition to one, should be done; in addition, make up one more order with three good rows at the same time, and exclude them ;......; Continue this process and get the number of wrong columns

M (n) = n! -N! /1! + N! /2! -N! /3! +... + (-1) ^ N * n! /N! = Sigma (k = 2 ~ N) (-1) ^ K * n! /K!

That is, M (n) = n! [1/0! -1/1! + 1/2! -1/3! + 1/4! +... + (-1) ^ n/n!]

Note: Sigma indicates the concatenation of symbols (k = 2 ~ N) is the connection range.

3. Simplified Formula

In addition, the formula for troubleshooting in the book is DN = n! (1/0! -1/1! + 1/2! -1/3! -... + (-1) ^ n/n !), This formula is inconvenient to calculate n when it is very large. later it was found that it can be reduced to a beautiful formula using series knowledge = [n! /E + 0.5], [x] is the integer function, that is, X is the rounded down.

The formula is relatively simple. Observe the formula in the general book, you can find that the E-1 before the same, and then compare/dN-n! E-1/<1/(n + 1) <0.5, so we get this simple and elegant formula (this is for reference only)